1 + 𝑖 𝑛 = 𝐹𝑉
𝑃𝑉
𝑛 ∙ 𝑙𝑛 1 + 𝑖 = 𝑙𝑛 𝐹𝑉
𝑃𝑉
𝒏 =
𝒍𝒏 𝑭𝑽
𝑷𝑽
𝒍𝒏 𝟏 + 𝒊We can solve for the
exponent using a rule
of (natural) logarithms
A​ non-interest bearing promissory note for ​$1263.00 was discounted at 6​% p.a. compounded monthly. If the proceeds of the note were ​$1071.00​, how long before the due date was the note​ discounted? State your answer in years and months​ (from 0 to 11​ months).
Question content area bottom
Part 1
The note was discounted

enter your response here ​year(s) and

enter your response here ​month(s) before the due date

To solve this problem, we'll use the formula for the present value (PV) of a future amount (FV) when interest is compounded monthly. The equation that relates the present value $ PV $, future value $ FV $, interest rate $ i $, and the number of periods $ n $ is:

\[
PV = \frac{FV}{(1 + i)^n}
\]

Where:
- $ FV $ = Future Value (\$1263.00)
- $ PV $ = Present Value (\$1071.00)
- $ i $ = Monthly interest rate
- $ n $ = Number of months (the value we need to find)

First, we need to determine the monthly interest rate. Since the annual interest rate is 6% compounded monthly:

\[
i = \frac{6\%}{12} = 0.5\% = \frac{0.5}{100} = 0.005
\]

Now, substitute the known values into the present value formula and solve for $ n $:

\[
1071 = \frac{1263}{(1 + 0.005)^n}
\]

To isolate $ n $, first divide both sides by 1263:

\[
\frac{1071}{1263} = \frac{1}{(1.005)^n}
\]

Simplify the fraction:

\[
0.847 = \frac{1}{(1.005)^n}
\]

Now, take the reciprocal of both sides:

\[
(1.005)^n = \frac{1}{0.847} \approx 1.1805
\]

To solve for $ n $, we take the natural logarithm (ln) of both sides:

\[
n \cdot \ln(1.005) = \ln(1.1805)
\]

Solve for $ n $:

\[
n = \frac{\ln(1.1805)}{\ln(1.005)}
\]

Now let's compute the value:

\[
n \approx \frac{0.1658}{0.004987} \approx 33.25
\]

So, $ n \approx 33.25 $ months.

To convert this to years and months:
- 33.25 months is approximately 2 years and 9 months.

Thus, the note was discounted **2 years and 9 months** before the due date.

Would you like further details or have any questions?

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Here are 8 related questions:
1. How does compounding frequency affect the present value?
2. What is the difference between simple interest and compound interest?
3. How would the answer change if the interest rate were compounded quarterly instead of monthly?
4. Can you calculate the future value of a note given the present value, interest rate, and time?
5. What happens to the present value if the interest rate increases?
6. How do we calculate the present value of an annuity with monthly payments?
7. What are the implications of discounting a note at a higher interest rate?
8. How does the formula change if the interest rate is compounded continuously?

**Tip:** Always convert annual interest rates to the relevant compounding period before using them in calculations to avoid errors.

Learn how to calculate the discounted value of a promissory note using the present value formula. This problem involves compound interest with monthly compounding, detailing steps to find the number of months before the due date when the note was discounted. Suitable for college-level finance and mathematics.

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